Because Jin Yong's wuxia novels *generally* (though not always in the details) follow established facts of history, the Sung Dynasty of China *had* to fall to the Mongols.
Setting that idea aside, however, the Greats logically *should* have been able to end the Mongol threat to Sung China by launching "decapitation attacks" to take out the Mongol leadership.
1. The Mongols were great warriors, but without effective leaders, they would not be able to overthrow a powerful established state such as the Sung Kingdom. In history, the outnumbered Mongol warriors were successful against seemingly more powerful enemies not only because the Mongols were disciplined, hardy, and determined, but also because they had intelligent and charismatic leaders. Without leaders such as Temujin, Jebe, Jochi, Ogodei, Chagahatai, Tolui, Mongke, Kublai, and others, the great Mongol Empire could not have come into being and been dominant throughout Eurasia for the better part of a century.
2. Wulin martial artists, no matter how powerful, cannot defeat entire armies in straight up combat, but as seen in DGSD (Kiu Fung, Hui Juk, and Deun Yu going around the Khitan army to reach Emperor Yeh Lut Hung Gei) and ROCH (Yeung Gor going through the Mongol army to kill Mongke Khan), it's not hard for Greats-or-higher level martial artists to go *around* armies to kill emperors, princes, and individual generals.
3. At the end of ROCH, wulin had five Greats and a few other fighters (e.g. Little Dragon Girl, Wong Yung) who were also very high-level.
4. The Mongols had lost their best wulin warrior (Golden Wheel Monk) and a few other powerful warriors (e.g. Lui Mor Singh, Wan Hak Sai, and Siu Seung Tze). Unless there were others Jin Yong never told us about, they didn't have much left in the way of wulin-style martial artists to oppose China's Greats.
Given all that, it seems improbable that the New Five Greats, having staggered the Mongols by killing Mongke, the Golden Wheel Monk, and the remaining Mongol mercenaries, didn't press their advantage by similarly killing off Kublai and any remaining Mongol princes and generals capable of posing a threat to Sung China. With their skills, the Greats should easily have been able to get around the Mongol army to assassinate such princes and generals. An army can tear a Great apart (though only with extreme difficulty and at a terrible cost in personnel lost) if the Great chooses to stand his ground and fight the army, but an army can't do much against a Great (let alone five of them) if the Great chooses to use stealth and hit-and-run tactics, which would be all a Great would need to assassinate a Mongol khan, prince, or general. If the Greats killed enough of these, they would soon degrade the Mongol leadership so badly that the Mongols could no longer pose a serious threat to Sung China.